We have a db that has a mission critical website sitting on top of it. We check it weekly with a DBCC CHECKDB command, which takes 1 to 2.5 minutes to run. Are web users likely to see issues during that time? If so, is there a better way to do a weekly health check on it?

Whats your RTO and RPO for the database? If its mission critical and DBCC CHECKDB only takes 1 - 2.5 minutes to run, I would run it every day not weekly. The only impact to the users would be while the snapshot is being created that DBCC CHECKDB runs on, so will be very minimal.

CheckDB is an online operation. Users may note slow down, but that's all.

Better question, if your database becomes corrupt, how soon do you want to find out about it and how long do you retain your backups? If you do find corruption, restore from backup should always be an option and that means doing integrity checks often enough that restore from a clean backup is always possible